Squaring the Square

A note on the dissection of a square into smaller squares all different except for a single pair of equal non-touching squares (or two such pairs)

Editor's Note: This article has been delayed in publication mainly because I was hoping to find a better way of presenting the dissection diagrams. Mr Willcocks provided A4-size diagrams prepared for him by Dr J. D. Skinner, but it was not possible to reproduce them adequately on the screen. However, George Bell (19 December 2004) has now been able to provide a scale drawing gif-image of the first dissection (A1) as shown above. The impossibility of showing an accurate diagram is evident from the fact that the square side is 6732 units and the screen width is only 800 pixels. So the above image, which is 600 pixels wide, compresses 11.22 units to a pixel. So squares smaller than this, as occur in B1, B3, C1, C2 below, would be lost. For the diagrams in the text I have used the conventional method of showing the squares as rectangles.

During the past 75 years much work has been done on the problem of dividing a rectangle into smaller squares all of different sizes. When the rectangle itself is a square the problem is known as Squaring the Square and a comprehensive theory was developed by four mathematicians at Cambridge by which they showed a relationship between the elements of dissections and the currents in certain electrical networks. Details may be found in [1] and [2]. A very interesting non-mathematical but authoritative account may be found in [3]. The following notes are based entirely on the electrical theory.

In the diagram of Fig.(1a) ABC represents a network where for simplicity interior wiring has been omitted. DEF represents the map which is the dual of an equivalent map to ABC (interior wiring again omitted). These two networks are joined by the wires BE, BD, DC, CF, and FA. If now a current enters the network at A and leaves at E, the theory tells us that (1) the network gives rise to a squared square and (2) the wires BD and CF carry zero currents.

Lying in my bath one evening with a mental picture of such a network, I realised that the wire XD could equally well be drawn as XB and similarly the wire YB could be drawn as YD. What I had done was to replace two sides of the quadrilateral BYDX with the other two sides. It seemed that the natural thing to do was to replace the diagonal BD with the other diagonal XY and see what resulted. I had no reason to expect any interesting outcome, but on evaluating the new network I discovered that (1) I had a squared square, (2) the currents in BX and DY were numerically equal, and (3) vectors could be assigned as shown in Fig.(1b). Knowledge of (2) and (3) can simplify the calculation of currents in the network.

Experiment showed that a similar procedure could be followed with the wire CF or with both wires with a view to obtaining a single pair of equal squares in two different ways or two pairs of equal squares in one way. I have constructed a considerable number of such squares without finding a failure, but I lack a strict proof justifying the procedure. As with all other methods of this kind it is important to check that there is no ‘unwanted’ imperfection.

An example of three squares constructed in this way showing an interesting arithmetical relationship is given in the accompanying diagrams C1, C2, C3. The sum of the duplicated squares in C1 and C2 (92 + 208) is equal to the sum of the two duplicated pairs in C3 (97 + 203).

This method is not restricted to the use of self-dual maps. A dissection of a rectangle in two different ways but with a corner square in one equal to a corner square in the other will yield two different tripolar networks and one of these can be used with the dual of the other.

In this way we can form four networks, and it is their close relationships with each other which result in some curious arithmetical equalities, such as that noted above. From A and B we have 431 + 187 = 332 + 286.

The writer does not claim that all initial networks give such interesting results. He has been content to show some that he has found.

A1: 34 squares, side 6732 units.

	3006							1870						1856
														651			1205
								1136			734
														97	554
											831
	1456	865			685			513		623
															374	1385
											194	1011
								403	110
									927
					454		231
		591	274
							634
			317			411
												110	2286
	2270			94
				2176

A2: 34 squares, side 6876 units.

	2637				2305									1934
														714			1220
					332		1422				551
	1834	1135
														208	506
											759
															461	1265
											416		804
							481	609		332
		699		436
											28	388
										360
				263		654
								526	83
									433			2457
	2405		391
			2014

A3: 35 squares, side 9447 units.

	3561				3130										2756
															1002			1754
					431		1821					878
	2476	1516
															250	752
												1128
																626	1880
							682	708	431
		960		556
												500	1254
									177		254
									100	77
				404		834
										831
								808
	3410		430
													77	3057
			2980

B1: 34 squares, side 6705 units.

	2592						1681								2432
							911				770
															202	2230
											141	446		183
	1285		1307				747			305
														385
										629			122
													507
							560		187
									1323
	1263	22
		939				950
																187	2043
		302		626	11
					615			1856
	1565
				1241

B2: 34 squares, side 6732 units.

	2432					2293								2007
														286	1721
						842				1737
	1870	562
		421		141
				374			669
		327	94
			233		235
															302		1419
		560
					558			286
								272	1102			751
	2430				830
												51	634			166
																1585
					1600						483
											1117

B3: 35 squares, side 4669 units.

	1804					1526									1339
															187		1152
						604					1109
	1490		314
			302		12
					298			318
			294	8
				286			20
							151			187
																	144	1008
							115		36
									774				702
			695
	1375	115
													72	468		162
		1260										396
																1170
												864

C1: 34 squares, side 6705 units.

	2817					2149									1739
															410		1329
						668				1048			843
	1555	777	485			288			380
													205	562		76
																1405
						196		92
								1368			357
						1	195
			292		194
					389
											919
		778		291
				680
	2333
											92	2232
				2140

C2: 34 squares, side 6876 units.

	2637							1762								2477
								875				887
																706	1771
	1175	644			818			513		362
												353	534
										359	3
											356
								305	208
													175	1065
		470		174
									1098
				296			1001
		61	705
	1236
														208	2628
			592			2420
	1828

C3: 35 squares, side 9413 units.

	3509			3306												2598
																708		1890
				203	1153						1506		1152
	2404	1308
													354		768		30
					358	380	212		203
																	1920
									53	150
							168	44
								97
								1693				414
		1096	570
						548
												1182
			1118
	3508
												97		3005
			2908

It is possible to make further speculations. It is obvious that the process is reversible. The method employed results in all cases with a square containing 3 component squares situated as in Fig.(2a). Reversing the process the two squares A are removed and a zero current is introduced.

Fig.2a A B A

Fig.2b A B A D A

What happens if we remove by this process in a square with a configuration as in Fig.(2b) the two squares A which adjoin square D? The small number of examples I have tested have all yielded squared squares.

The sceptical reader might care to test the inverse method with the following square, Fig.(3). At first sight this does not look like a candidate but it can be made one by treating the square [4] as being 4 squares of side 2 surrounding a zero square. To my surprise I found that this resulted in a square (imperfect of side 313). Is it conceivable that a perfect square could be found in this way?

Fig.(3). 21 squares, side 107

	34		36				37
	32	2
		21			17
							16			21
					4	20			9
		9		16
										4	17
									13
	41
				34				2
								32

In conclusion I would like to express my thanks to Dr J. D. Skinner for his help with the diagrams.

References.
[1] The dissection of rectangles into squares R. L. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, (Duke Math J. Vol.7 p.312-340)
[2] Squaring the Square W. T. Tutte (Can. J. Math. Vol.2 p.197-209, 1950).
[3] Squaring the Square W. T. Tutte (Chapter 17 of the 2nd Scientific American Book of Mathematical Puzzles and Diversions).

George Bell cites the following website on squaring the square, though I've not been able to access it properly since its "javascript" does not pass my computer's security settings. www.squaring.net

Update (2 June 2013): My more recent computers have had no trouble with access to the site. I have corresponded with the author Stuart Anderson. There is now a page devoted to T. H. Willcocks and his work: THW on www.squaring.net. Mr Willcocks reached the age of 100 on 19 April 2012.